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BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI Publicat de

Universitatea Tehnică „Gheorghe Asachi” din Iaşi Tomul LIX (LXIII), Fasc. 3, 2013

Secţia CONSTRUCŢII. ARHITECTURĂ

INFLUENCE OF LOADING PARAMETERS ON LATERAL TORSIONAL BUCKLING RESISTANCE

OF STEEL PLATE GIRDERS

BY

PETRU MOGA, DELIA DRĂGAN and CLAUDIA ALB*

Technical University of Cluj-Napoca Faculty of Civil Engineering

Received: May 27, 2013 Accepted for publication: June 11, 2013

Abstract. The results of an analysis regarding the influence that the load

location relative to shear centre of the section and the cross-section shape have on the bending resistance of the mono-symmetrical steel plate girders are presented. The girder bending resistance is evaluated according to the Euro norms EN 1993-1-1: Design of Steel Structures. General Rules and Rules for Buildings and EN 1993-1-5: Plated Structural Elements.The obtained results can be useful in the optimal design of steel plate girders.

Key words: steel plate girders; lateral-torsional buckling; bending resistance; influence parameters; Euronorms EN 1993-1-1: Design of Steel Structures. General rules and rules for buildings; EN 1993-1-5: Plated Structural Elements.

1. Introduction

The steel girders subjected to bending generally have a very different

rigidity relative to the main axes (Iy >> Iz) so, they should be verified against lateral-torsional buckling. *Corresponding author: email: [email protected]

148 Petru Moga, Delia Drăgan and Claudia Alb

In the most practical cases of steel buildings and bridges, girders are built up of hot-rolled I and H sections or welded sections.

The evaluation of the actual critical bending moment for each loading pattern leads to solving a complex of differential eqs. so, in the design activity we use conservative and approximate eqs., useful in the most practical cases.

This paper presents an analysis regarding the influence that the load location relative to shear centre of the section and the cross-section shape have on the lateral torsional buckling resistance of the mono-symmetrical steel plate girders. The girder bending resistance is evaluated according to the Euronorms EN 1993-1-1: Design of Steel Structures. General Rules and Rules for Buildings and EN 1993-1-5: Plated Structural Elements.

2. Lateral-Torsional Buckling Resistance

2.1. Buckling Resistance

According to EN 1993-1-1:2006 - §6.3.2, the design buckling resistance

moment of a laterally unrestrained beam should be taken as:

LT1

yb.Rd y

M

fM W

, (1)

where:

, for class1or 2 cross-sections;, for class 3 cross-sections;

, for class 4 cross-sections;

pl .y

y el .y

eff .y

WW W

W

(2)

ΧLT is the reduction factor for lateral-torsional buckling. For bending moments of constant cross-sections, the value ΧLT for the

appropriate non dimensional slenderness, LT , should be determined from:

LT LT22LTLT LT

1 , but 1,

(3)

where: 2LT LTLT LT0 5 1 0 2. .

; αLT is an imperfection factor;

LT cry yW f M ; Mcr is the elastic critical moment for lateral-torsional buckling.

Bul. Inst. Polit. Iaşi, t. LIX (LXIII), f. 3, 2013 149

The imperfection factor, αLT , corresponding to the appropriate buckling curve may be obtained from Table 1.

Table 1 Imperfection Factors

Buckling curve a b c d Imperfection factor, αLT 0.21 0.34 0.49 0.76

The recommendations for buckling curves are given in Table 2.

Table 2. Buckling Curves

Cross-section Limits Buckling curve

Rolled I - section h/b ≤ 2 a h/b > 2 b

Welded I - section h/b ≤ 2 c h/b > 2 d

Other cross - section – d

2.2. Elastic Critical Bending Moment

For a double T cross-section (I-section), the elastic critical moment for lateral torsional buckling is given by the expression derived from the buckling theory

222 2

cr 1 2 3 2 32 2twz

g j g jw z z

kL GIIEI kM C C z C z C z C z .k I EIkL g

(4)

For doubly-symmetric I-section, zj = 0, and the expression (4) becomes

222 2

cr 1 2 22 2twz

g gw z z

kL GIIEI kM C C z C z .k I EIkL g

(5)

where: C1, C2 and C3 are coefficients depending on the loading (moment diagram), section properties and support conditions,

1 1 0z

y

Ig . .I

If there are no destabilizing loads, zg = 0 and it follows:


222

cr 1 2 2twz

w z z

kL GIIEI kM C k I EIkL

. (6)

3. Analysis of Loading Parameters on Buckling Resistance

In what follows next, we analyse the influence of the loading parameters and of the cross-section shape on the lateral torsional buckling resistance of a girder with a mono-symmetric section.

The following analytical data are known: a) static scheme, loading and the cross-section, Fig. 1; b) material: steel S 355 – fy = 355 N/mm2; c) the girder is laterally restrained at the end supports. The following cases are analyzed: a) the compression flange is larger in comparison with the tension

flange (Iz is greater) ; b) the compression flange is smaller in comparison with the tension

flange (Iz is smaller); The force acts in the following points of the cross-section: a) at the compression flange (with a destabilizing effect); b) in the shear centre; c) at the tension flange.

Fig. 1 – Design data.


Solution: In Tables 3 a and 3 b we present the analyzed cases (a total of 6 cases)

and the geometric design parameters used to evaluate the lateral buckling resistance.

Table 3 a Loading Parameters

Table 3 b Loading Parameters


3.1. Cross-Section Class

C a s e s 1: The compression flange is greater. Compression flange: ψ = 1, kσ = 0.43,

300 12 2

5 76 9 7 29 class 125/c . . .t

The web is an internal, partially compressed plate, Fig. 2 a.

22 1 1 1

1

488 1 56 1 56 1,312z . .z , 312 0 39 0 5800 . . ,

800 66 67 36 74 77 class112c . / . .t

It results: that cross section belongs to class 1.

Fig. 2 – Evaluation of Cross – section class.

C a s e s 2: The compression flange is smaller

Compression flange:

160 12 23 7 9 7 29 class120

/c . .t

Web (see Fig. 2 b):


22 1 1 1

1

312 0 64 0 64 1488z . .z ,

488 0 61 0 5800 . . ,

800 66 6712c .t

456 4253 29; 74 15 class 213 1 0 67 0 33c c. .t t . .

.

It results: that cross section belongs to class 2 Parameters valid for all cases: αLT = 0.76 – curve d for h/b > 2 (Table 1

and Table 2);

2 2 62

cr 2 2

2 1 10 6,31910 3,634 33 kN600

z.z

EI .N .kL

,

600 107 14 ,5 6zz

kL .i . 1 93 9 93 9 0 81 76 1,y

E . . . .f

pl1 0y

w.y

W.W ,

1 2 31 365, 0 553, 1 73C . C . C . ,

1 cr 1 365 3,634 33 4,961 kN.zC N . . ,

66 64 13 10 653 58; 0 81 10 245 198 45 106,319

wt

z

I . . GI . .I

,

3pl 5,941cm.yW .

The coefficient zj

0 4 2 1 , for 0 5,

0 5 2 1 , for 0 5,

s f f

js f f

. h .z

. h .

where: 3

3 3fc fc

ffc fc ft ft

b tb t b t

;

zj is positive when the flange with greater Iz is in the compression zone.


C a s e 1 3 3

3 3 3 3

30 2 5 0 89 0 5,30 2 5 16 2

0 4 82 25 (2 0 89 1) 25 66.

fc fcf

fc fc ft ft

j

b t . . .b t b t .

z . . . .

C a s e 2 3 3

3 3 3 3

16 2 0 108 0 5,30 2 5 16 2

0 5 82 25 (2 0 108 1) 32 24.

fc fcf

fc fc ft ft

j

b t. .

b t b t .

z . . . .

3.2. Elastic Critical Buckling Moments

By using the expression (4 a), the following cases may be considered: C a s e 1A

62 2

cr

4

198 45 104,961 10 653 58 (0 553 10 2 1 73 25 66)496,100

(0 553 10 2 1 73×25 66) 10 4,430 kN.m.

.M . . . . .

. . . .

C a s e 1B

62 2

cr

4

198 45 104,961 10 653 58 ( 1 73 25 66)496,100

( 1 73 25 66) 10 4,930 kN.m.

.M . . .

. .

C a s e 1C

622

cr

4

198 45 104,961 10 653 58 0 553 ( 74 3) 1 73 25 66496,100

0 553 ( 74 3) 1 73 25 66 10 8,777 kN.m.

.M . . . . .

. . . .


C a s e 2A

622

cr

4

198 45 104,961 10 653 58 0 553 74 3 1 73 ( 32 24)496,100

0 553 74 3 1 73 ( 32 24) 10 263 kN.m.

.M . . . . .

. . . .

C a s e 2B

622

cr

4

198 45 104,961 10 653 58 1 73 ( 32 24)496,100

1 73 ( 32 24) 10 263 kN.m.

.M . . .

. .

C a s e 2C

622

cr

4

198 45 104,961 10 653 58 0 553 ( 10 2) 1 73 ( 32 24)496,100

0 553 ( 74 3) 1 73 25 66 10 477 kN.m.

.M . . . . .

. . . .

3.3. Non-Dimensional Slenderness, LT , and Reduction Factors, LT The non-dimensional slenderness is evaluated with the expression:

LTy y

cr

W fM , where: 3

pl 5,941cmy .yW W , pl 2,109 kN.m.y yW f .

Taking into account the values of the non dimensional slenderness, LT , obtained after the calculation of the critical moments, Mcr , the reduction factors, LT , can be evaluated

LT 22LTLT LT

1

,

where: 2LT LTLT LT0 5 1 0 2. .

, αLT = 0.76 – curve d for h/b >2

(Tables 1 and 2).


The following values are obtained:

C a s e 1A

LT2,109 0 694,430 . ; 2

LT 0 5 1 0 76 0 69 0 2 0 69 0 92. . . . . . ;

LT 2 2

1 0 650 92 0 92 0 69

.. . .

.

C a s e 1B

LT2,109 0 654,930 . ; 2

LT 0 5 1 0 76 0 65 0 2 0 65 0 88. . . . . . ;

LT 2 2

1 0 680 88 0 88 0 65

.. . .

.

C a s e 1C

LT2,109 0 498,777 . ; 2

LT 0 5 1 0 76 0 49 0 2 0 49 0 73. . . . . . ;

LT 2 2

1 0 770 73 0 73 0 49

.. . .

.

C a s e 2A

LT2,109 2 83263 . ; 2

LT 0 5 1 0 76 2 83 0 2 2 83 5 50. . . . . . ;

LT 2 2

1 0 105 5 5 5 2 83

.. . .

.

C a s e 2B

LT2,109 2 20434 . ; 2

LT 0 5 1 0 76 2 20 0 2 2 20 3 68. . . . . . ;


LT 2 2

1 0 153 68 3 68 2 2

.. . .

C a s e 2C

LT2,109 2 10477 . ; 2

LT 0 5 1 0 76 2 10 0 2 2 10 3 43. . . . . . ;

LT 2 2

1 0 163 43 3 43 2 1

.. . .

.

3.4. Bending Moment Resistance

According to EN 1993-1-1:2006, §6.3.2, the design buckling resistance

moment of a beam should be evaluated with the expression (2).

4. Final Remarks and Conclusions

The analysis results are presented in Table 4.

Table 4 Parameters LT and LT

Case Parmeter

LT LT 1A 0.69 0.65 1B 0.65 0.68 1C 0.49 0.77 2A 2.83 0.10 2B 2.20 0.15 2C 2.10 0.16

Table 5 synthetically presents the analysis results. Taking into account that the cross-section class is 1 or 2, the ratio of the

bending moment is equal with the ratio of the reduction factors ( ply .yW W ). It can be observed that the maximum buckling moment resistance is

obtained when the load force has a stabilizing effect and the compression flange is larger in comparison with the tension flange.


Table 5

minb.Rd b.RdM M Ratio

Case Scheme minb.Rd b.RdM M Case Scheme min

b.Rd b.RdM M

1A

6.5 2A

1.0

1B

6.8 2B

1.5

1C

7.7 2C

1.6


REFERENCES

* * * Design of Steel Structures. Part 1-1: General Rules and Rules for Buildings. SR EN

1993-1-1/2006, Eurocode 3. * * * Design of Steel Structures. Part 1-5: Plated Structural Elements. SR EN 1993-1-

5/2008 , Eurocode 3. * * * Verificarea la stabilitate a elementelor din oţel în conformitate cu SR EN 1993-1.1.

Recomandări de calcul, comentarii si exemple de aplicare. Contract nr. 424/08.12.2009, Timişoara, 2010.

* * * NCCI: Elastic Critical Moment for Lateral Torsional Buckling. SN003a – EA – EU, 2006. www.access-steel.com.

* * * Cold Formed Gauge Members and Sheeting, Seminar on Eurocode 3. Part 1.3, Tempus 4502-94, Timişoara, 1995.

* * * Design of Steel Structures. Seminar on Eurocode 3, Tempus 4502-92, Timişoara. 1993

Moga P., Grinzi metalice zvelte. Univ. Tehn., Cluj-Napoca, 2011

INFLUENŢA PARAMETRILOR DE ÎNCĂRCARE ASUPRA REZISTENŢEI LA INCOVOIERE CU FLAMBAJ LATERAL, A GRINZILOR METALICE

(Rezumat)

Se prezintă rezultatele unei analize privind influenţa poziţiei încărcării în raport

cu centrul de răsucire – forfecare al secţiunii transversale şi influenţa formei secţiunii asupra rezistenţei la încovoiere cu flambaj lateral a grinzilor metalice cu secţiune monosimetrică. Calculul momentelor capabile se efectuează în conformitate cu norma SR EN 1993-1-1:2006: Proiectarea structurilor de oţel. Partea 1-1: Reguli generale şi reguli pentru clădiri.

Rezultatele obţinute pot fi utile în activitatea de proiectare optimală a grinzilor cu inimă plină care sunt susceptibile de a suferi fenomenul de pierdere a stabilităţii generale prin flambaj lateral.

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